“order”的概念、定义、翻译、参考文献-科学参考

order

Physics

1. The number of times a variable is differentiated. dy/dx represents a first-order derivative, d²y/dx² a second-order derivative, etc. In a differential equation the order of the highest derivative is the order of the equation. d²y/dx²+2dy/dx=0 is a second-order equation of the first degree. See also degree.

2. (in physics) A category of phase transition.

3. The arrangement a system has below a transition point. See order parameter.

Mathematics

The order < on the real numbers satisfies the following axioms; here x, y, z denote real numbers:
1. (i) (Trichotomy) Precisely one of x < y, y < x, x = y holds.
2. (ii) (Transitivity) If x < y and y < z, then x < z.
3. (iii) If x < y, then x + z < y + z.
4. (iv) If 0<z and x < y, then zx < zy.
Together with the field axioms, (i)–(iv) above, state ℝ is an ordered field. Other examples are the rational numbers and the hyperreals. The real numbers also satisfy the completeness axiom and are the unique complete ordered field.
See ordinal number, partial order, total order, well ordered.

Chemistry

In the expression for the rate of a chemical reaction, the sum of the powers of the concentrations is the overall order of the reaction. For instance, in a reaction
$A + B \to C$
the rate equation may have the form
$R = k {[A] [B]}^{2}$
This reaction would be described as first order in A and second order in B. The overall order is three. The order of a reaction depends on the mechanism and it is possible for the rate to be independent of concentration (zero order) or for the order to be a fraction. See also molecularity; pseudo order.

Computer

1. A means of indicating the way a function varies in magnitude as its argument tends to some limits, usually zero or infinity. More precisely if there is some constant K such that
$| f (x) | \leq K ϕ (x)$
for all x ≥ x₁, then we say that f(x) is order ϕ‎(x) as x tends to infinity, and we write
$f (x) = O (ϕ (x))$
For example,
$\begin{array}{l} 100 x^{2} + 100 x + 2 = O (x^{2}) \\ as x \to \infty \end{array}$
If
$lim_{x \to a} f (x) / g (x) = 0$
then we write
$f (x) = o (g (x))$
For example,
$x = o (x^{2}) as x \to \infty$
Both these notations are statements about maximum magnitude and do not exclude f from being of smaller magnitude. For example,
$x = O (x^{2})$
is perfectly valid, but equally
$x = O (x)$
If
$lim_{x \to a} f (x) / g (x) = const . k \neq 0$
then we write
$f (x) ≅ k g (x) as x \to a$
For example,
$\begin{array}{l} 10 x^{2} + x + 1 ≅ 10 x^{2} \\ as x \to \infty \end{array}$
The term order and the O notation is used in numerical analysis, particularly in discretization methods. In ordinary differential equations, if h denotes the stepsize, then a method (or formula) has order p (a positive integer) if the global discretization error is O(h^p). This means that as the step size h is decreased, the error goes to zero at least as rapidly as h^p. Similar considerations apply to partial differential equations. High-accuracy formulas (order up to 12 or 13) are sometimes used in methods for ordinary differential equations. For reasons of computational cost and stability, low-order formulas tend to be used in methods for partial differential equations.
The term is also used to refer to the speed of convergence of iteration schemes, for example Newton’s method for computing the zero of a function f(x). Subject to appropriate conditions, Newton’s method converges quadratically (or has order of convergence 2), i.e. an approximate squaring of the error is obtained in each iteration.
2. Another name for operation code.

Geology and Earth Sciences

See classification.

Philosophy

See first-order language.

单词	order
释义	order Physics 1. The number of times a variable is differentiated. dy/dx represents a first-order derivative, d²y/dx² a second-order derivative, etc. In a differential equation the order of the highest derivative is the order of the equation. d²y/dx²+2dy/dx=0 is a second-order equation of the first degree. See also degree. 2. (in physics) A category of phase transition. 3. The arrangement a system has below a transition point. See order parameter. Mathematics The order < on the real numbers satisfies the following axioms; here x, y, z denote real numbers: (i) (Trichotomy) Precisely one of x < y, y < x, x = y holds. (ii) (Transitivity) If x < y and y < z, then x < z. (iii) If x < y, then x + z < y + z. (iv) If 0<z and x < y, then zx < zy. Together with the field axioms, (i)–(iv) above, state ℝ is an ordered field. Other examples are the rational numbers and the hyperreals. The real numbers also satisfy the completeness axiom and are the unique complete ordered field. See ordinal number, partial order, total order, well ordered. Chemistry In the expression for the rate of a chemical reaction, the sum of the powers of the concentrations is the overall order of the reaction. For instance, in a reaction $A + B \to C$ the rate equation may have the form $R = k {[A] [B]}^{2}$ This reaction would be described as first order in A and second order in B. The overall order is three. The order of a reaction depends on the mechanism and it is possible for the rate to be independent of concentration (zero order) or for the order to be a fraction. See also molecularity; pseudo order. Computer 1. A means of indicating the way a function varies in magnitude as its argument tends to some limits, usually zero or infinity. More precisely if there is some constant K such that $\| f (x) \| \leq K ϕ (x)$ for all x ≥ x₁, then we say that f(x) is order ϕ‎(x) as x tends to infinity, and we write $f (x) = O (ϕ (x))$ For example, $\begin{array}{l} 100 x^{2} + 100 x + 2 = O (x^{2}) \\ as x \to \infty \end{array}$ If $lim_{x \to a} f (x) / g (x) = 0$ then we write $f (x) = o (g (x))$ For example, $x = o (x^{2}) as x \to \infty$ Both these notations are statements about maximum magnitude and do not exclude f from being of smaller magnitude. For example, $x = O (x^{2})$ is perfectly valid, but equally $x = O (x)$ If $lim_{x \to a} f (x) / g (x) = const . k \neq 0$ then we write $f (x) ≅ k g (x) as x \to a$ For example, $\begin{array}{l} 10 x^{2} + x + 1 ≅ 10 x^{2} \\ as x \to \infty \end{array}$ The term order and the O notation is used in numerical analysis, particularly in discretization methods. In ordinary differential equations, if h denotes the stepsize, then a method (or formula) has order p (a positive integer) if the global discretization error is O(h^p). This means that as the step size h is decreased, the error goes to zero at least as rapidly as h^p. Similar considerations apply to partial differential equations. High-accuracy formulas (order up to 12 or 13) are sometimes used in methods for ordinary differential equations. For reasons of computational cost and stability, low-order formulas tend to be used in methods for partial differential equations. The term is also used to refer to the speed of convergence of iteration schemes, for example Newton’s method for computing the zero of a function f(x). Subject to appropriate conditions, Newton’s method converges quadratically (or has order of convergence 2), i.e. an approximate squaring of the error is obtained in each iteration. 2. Another name for operation code. Geology and Earth Sciences See classification. Philosophy See first-order language.
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